Approximating the set of separable states
using the positive partial transpose test
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Citation Beigi, Salman, and Peter W. Shor. “Approximating the Set of
Separable States Using the Positive Partial Transpose Test.” Journal of Mathematical Physics 51.4 (2010): 042202. Web.
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arXiv:0902.1806v2 [quant-ph] 21 Jun 2011
Approximating the Set of Separable States Using the Positive
Partial Transpose Test
Salman Beigi1 and Peter W. Shor2 1
Institute for Quantum Information,
California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: June 22, 2011)
Abstract
The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, are efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion and symmetric extension criterion. We also bring an evidence that the sets of positive partial transpose states and separable states have totally different shapes.
I. INTRODUCTION
The problem of detecting entanglement has been focused in quantum information theory for many years. The problem is: given a bipartite mixed state ρAB, decide whether this
state is entangled or separable. The first attack toward solving this problem is the following observation due to Peres and the Horodeckis [1, 2]. If ρAB = Pipiρ(i)A ⊗ ρ
(i)
B is separable,
then (ρAB)TB = Pipiρ(i)A ⊗ (ρ (i)
B)T, where MT denotes the transpose of matrix M, is also
a quantum state, and is a positive semi-definite matrix. Therefore, if ρAB is separable, its
partial transpose, (ρAB)TB, should be positive semi-definite. The Horodeckis have proved
that this criterion characterizes all separable states in dimensions 2×2 and 2×3 [2]. However, there are entangled states in dimension 3 × 3 with a positive partial transpose [3].
Although the set of positive partial transpose states (PPT states) does not coincide with the set of separable states, it is usually considered as an approximation of this set. For example in [4] instead of estimating the distance from separable states, the distance of an arbitrary state from PPT states has been computed as an “strongly related problem.” Also in [5] the geometry of the set of PPT states has been studied to understand the properties of the set of separable states. However, we do not know how efficient these approximations are. For instance, given an upper bound on the distance of a state from PPT states, does it give an upper bound on the distance from separable states?
We can think of this problem from the point of view of complexity theory. Gurvits [6] has proved that given a bipartite density matrix ρAB, it is NP-hard to decide whether this
state is separable or entangled. An approximate formulation of this problem is the following: given a bipartite density matrix ρAB and ǫ > 0, decide whether there exists a separable state
in the ǫ-neighborhood (in trace distance) of ρAB. Gurvits has established a reduction from
Knapsack to this problem, and has proved the NP-hardness of the separability problem, but only for exponentially small ǫ. However, as mentioned in [7] by replacing Knapsack with 2-out-of-4-SAT and repeating a similar argument, the NP-hardness can be proved for an inverse polynomial ǫ. Also, Gharibian [8] has shown the same result using a reduction from the Clique problem. Now the question is that how large ǫ can be while getting to the NP-hardness. For example, is there an efficient algorithm to decide whether the distance of a given state from separable states is less than 1/3, or it is an NP-hard problem? Equivalently, is there an efficiently implementable separability test such that if a state passes the test, then it is 1/3-close to the set of separable states?
In this paper we consider the converse of this question, i.e. given a separability criterion, if a state passes this test, can we claim a non-trivial upper bound on the distance of this state from separable states? We prove that the answer for the PPT criterion, as well as other well-known separability tests such as reduction criterion [9], majorization criterion [10], and symmetric extension criterion [11, 12], is no. More precisely, we prove the following theorem.
Theorem 1 Let H be a bipartite Hilbert space. For every ε > 0, if the dimension of each subsystem of H is large enough, there exists a PPT state acting on H whose trace distance from separable states is at least 1 − ε.
To the best knowledge of authors, this is the first result that compares separable states relative to PPT states in terms of their distance. However, the volume of these sets has been studied by several authors. For instance, by estimating the volume of separable states and PPT states in the Hilbert-Schmidt norm, it has been shown in [13] that a random PPT
state is entangled. The same conclusion has been proved in [14] in terms of Bures volume. See also [15] and [16] for some other results in this setting.
A. Main ideas
Let H = HA⊗ HB be a bipartite Hilbert space. We want to find PPT states ρ(n) ∈ H⊗n
such that the trace distance of ρ(n) from separable states is close to 1, for enough large
numbers n. Suppose ρ is an entangled PPT state. Then ρ⊗n is entangled and also PPT.
We claim that the sequence of states ρ(n) = ρ⊗n works for us. The intuition is that for two
different quantum states ρ and σ, the trace distance of ρ⊗n and σ⊗n tends to 1 as n tends
to infinity. However, in this problem σ is not a fixed state and ranges over all separable states. Also, it is not obvious (and may not hold) that the closest separable state to ρ⊗n
is of the form σ⊗n. (If we replace the trace distance with E
R(ρ), the relative entropy of
entanglement, this property does not hold [17].)
Another idea is to use entanglement distillation. Suppose the state ρ is distillable. It means that, having arbitrary many copies of ρ, using local quantum operations and classical communications (LOCC maps), we can obtain as many EPR pairs as we want (say m pairs). LOCC maps send separable states to separable states, and the trace distance decreases under trace preserving quantum operations. Therefore, the distance of ρ⊗n from separable states
is bounded from below by the distance of EPR⊗m from separable states, which we know is
close to 1 for large numbers m. Therefore, if ρ is distillable, the trace distance of ρ⊗n from
separable states tends to 1 as n tends to infinity.
It is well-known that PPT states are not distillable under LOCC maps. So we cannot use this idea directly. On the other hand, in this argument, the only property of LOCC maps that we use, is that they send separable states to separable states. Thus we may replace LOCC maps with non-entangling maps, the maps that send every separable state to a separable state. Due to the seminal work of Brandao and Plenio [18, 19] every entangled state is distillable under asymptotically non-entangling maps. As a result, by replacing LOCC maps with asymptotically non-entangling maps and repeating the previous argument, we conclude that the trace distance of ρ⊗n from separable states tends to 1.
Although this idea gives a full proof of Theorem 1, we do not present it in this paper. In-stead, we use more fundamental techniques, namely, quantum state tomography and quantum de Finetti theorem [20, 21]. In fact, these two techniques are the basic ideas of the results of [18, 19] that we mentioned above. Since ρ⊗(n+k) is a symmetric state, we may assume that
the closest separable state to ρ⊗(n+k) is also symmetric. Then by tracing out k registers and
using the finite quantum de Finetti theorem we conclude that the trace distance of ρ⊗(n+k)
from separable states is lower bounded by the trace distance of ρ⊗n from separable states of
the form X
i
piσi⊗n. (1)
Since such a state is separable and ρ is entangled, the sum of pi’s for which σi is close to
ρ cannot be large. On the other, if σi is far from ρ, using quantum state tomography one
can distinguish ρ⊗n from σ⊗n
i . Putting these two points together we show that the trace
distance of ρ⊗n and a separable state of the form of (1) is close to 1 for large enough n.
Note that in both of these arguments the only property of PPT states that we use, is that if ρ and σ are PPT, then ρ ⊗ σ is also PPT. So we can conclude the same result for any separability test which satisfies this property.
II. PRELIMINARIES
A pure state |ψi ∈ HA⊗ HB is called separable if it can be written of the form |ψi =
|ψAi ⊗ |ψBi, where |ψAi ∈ HA and |ψBi ∈ HB. A density matrix acting on HA ⊗ HB is
called separable if it can be written as a convex combination of separable pure states |ψihψ|. We denote the set of separable states by SEP.
For two quantum states ρ and σ we denote their trace distance by
kρ − σktr=
1
2tr|ρ − σ|, (2)
where |X| =√X†X.
A. Separability tests
Assume that dim HA = dim HB = d, and fix an orthonormal basis |1i, . . . , |di for both
of Hilbert spaces. The partial transpose of matrices acting on HA⊗ HB is a linear map
defined by (MA⊗ NB)TB = MA⊗ NBT, where the transpose is taken with respect to the fixed
basis. Clearly, if ρAB is a separable state, ρTABB is also a density matrix and then positive
semi-definite. However, it does not hold for an arbitrary state. For example, the partial transpose of the maximally entangled state is not positive semi-definite; let Φ(d) to be the maximally entangled state on H
Φ(d) = 1 d d X i,j=1 |i, iihj, j|. (3)
Φ(d) is not positive semi-definite because
Φ(d)TB = 1 d X i,j |iihj| ⊗ |jihi| = 1 dI − 1 d X i6=j |iihi| ⊗ |jihj| +d1X i6=j |iihj| ⊗ |jihi| = 1 dI − 2 d X i<j |φijihφij|, where |φiji = 1 √ 2(|ii|ji − |ji|ii). (4) As a result, positive partial transpose is a test to detect entanglement [1, 2]. More formally, if we denote the set of density matrices with a positive semi-definite partial transpose by PPT, then SEP ⊆ PPT.
Here is a list of some other separability criteria (see [22, 23]).
• Reduction criterion [9]: I ⊗ ρB ≥ ρAB, where ρB = trA(ρAB). Here, by M ≥ N we
• Entropic criterion [24]: Sα(ρAB) ≥ Sα(ρA) for α = 2 and in the limit α → 1, where
Sα(ρ) = 1−α1 log tr(ρα).
• Majorization criterion [10]: λ↓
ρA ≻ λ↓ρAB, where λ↓ρ is the list of eigenvalues of ρ in
non-increasing order, and y ≻ x means that, for any k, the sum of the first k entries of list x is less than or equal to that of list y.
• Cross norm criterion [25, 26]: tr|U(ρAB)| ≤ 1, where U is a linear map defined by
U(M ⊗N) = v(M)v(N)T and v(X) = (col
1(X)T, . . . , cold(X)T)T, where coli(X) is the
i-th column of X.
All of these tests for separability are necessary conditions but not sufficient. Doherty et al. [11, 12] have introduced a hierarchy of separability criteria which are both necessary and sufficient. Let ρAB =Pipiσi⊗ τi be a separable state. Then
ρAB1B2···Bk =X
i
piσi⊗ τi⊗k
is an extension of ρAB, meaning that ρAB = trB2···Bk(ρAB1···Bk). Also it is symmetric, meaning
that it does not change under any permutation of subsystems Bi. More precisely, for any
permutation π of k objects, if we define the linear map Pπ by Pπ|ψ1i ⊗ · · · ⊗ |ψki = |ψπ(1)i ⊗
· · · ⊗ |ψπ(k)i, we have
PB1...Bk
π ρAB1B2···Bk(PπB1...Bk)† = ρAB1B2···Bk. (5)
If such an extension exists, we say that ρAB has a symmetric extension to k copies. Doherty
el al. have proved that a quantum state is separable if and only if it has a symmetric extension to k copies for any number k [11, 12]. Also, they have shown that the problem of checking whether a given state has a symmetric extension to k copies, for a fixed k, can be expressed as a semi-definite programming, and can be solved efficiently (however, the size of this semi-definite program grows exponentially in terms of k). So we get to another separability test.
• Symmetric extension criterion [11, 12]: if ρAB is separable, then it has a symmetric
extension to k copies.
B. Quantum state tomography
An informationally complete POVM on H is a set of positive semi-definite operators {Mn}
forming a basis for the space of hermitian matrices on H, and such thatPnMn = I. In [21]
there is an explicit construction of an informationally complete POVM in any dimension. Such a POVM is useful for quantum state tomography.
Suppose {M∗
n} is the dual of basis {Mn}, i.e. tr(MnMm∗) = δmn, where δmn is the
Kronecker delta function. For any hermitian operator X we have
X =X
n
Therefore, having some copies of the state ρ, by measuring ρ using the POVM {Mn}, we
can approximate tr(ρMn) and then find the matrix representation of ρ.
Assume that H = HA⊗ HB is a bipartite Hilbert space. If {MA
n} and {MmB} are
infor-mationally complete POVM’s on HA and HB respectively, then {MA
n ⊗ MmB} is an
informa-tionally complete POVM on H. This means that, if the state ρAB is shared between two far
apart parties, they can perform quantum state tomography using classical communication. As a result, if the state ρAB is separable, then all intermediate states during the process are
separable as well.
C. Quantum de Finetti theorem
As in (5), a quantum state ρ(n) acting on H⊗n is called symmetric if P
πρ(n)Pπ† = ρ(n)
for any permutation π of n objects. A symmetric state is called k-exchangeable if it has a symmetric extension to n+k registers, i.e. a symmetric state ρ(n+k) such that tr
1,...,kρ(n+k) =
ρ(n). Clearly, any state of the form ρ⊗n is k-exchangeable, for any k. Also any convex
combination of these states is k-exchangeable. Quantum de Finetti theorem says that the converse of this observation holds: if a state is k-exchangeable for every k, it is in the convex hall of symmetric product states.
Quantum de Finetti theorem gives a characterization of infinitely-exchangeable states. The following theorem, known as the finite quantum de Finetti theorem, says that if a state is k-exchangeable (but not necessarily (k + 1)-exchangeable), then an approximation of the above result holds.
Theorem 2 [20] Assume that ρ(n+k) is a symmetric state acting on H⊗n+k. Let ρ(n) =
tr1...kρ(n+k) be the state obtained by tracing out the first k registers. Then there exists a
probability measure µ on the set of density matrices on H such that kρ(n)−
Z
µ(dσ)σ⊗nktr ≤ 2 dim H
n n + k.
III. PROOF OF THEOREM 1
As we mentioned our proof is based on the work of Brandao and Plenio about reversibility of entanglement transformation under asymptotically non-entangling maps. In particular, we follow similar steps as in the proof of Corollary II.2 of [27].
Let H = HA ⊗ HB and assume that d = dim H > 6. Then there exists a PPT state
ρAB = ρ acting on H which is not separable (see [3]). Let
ǫ = min
σ∈SEPkρ − σktr. (6)
Since ρ is entangled, ǫ > 0.
For every number n, ρ⊗ncan be considered as a bipartite state acting on (HA)⊗n⊗(HB)⊗n,
and it is a PPT state. Therefore, if we prove that the trace distance of ρ⊗n from separable
states tends to 1, as n goes to infinity, we are done.
permutation π we have
kρ⊗n− P
πσ(n)Pπ†ktr= kρ⊗n− σ(n)ktr,
and by triangle inequality
kρ⊗n−n!1 X π Pπσ(n)Pπ†ktr ≤ 1 n! X π kρ⊗n− Pπσ(n)Pπ†ktr = kρ⊗n− σ(n)ktr.
Therefore, we may assume that σ(n) is symmetric.
Let σ(n+n2)
be the closest (symmetric) separable state to ρ⊗(n+n2)
, and let tr1...n2σ(n+n 2)
be the state obtained by tracing out n2 registers. We have
kρ⊗(n+n2)− σ(n+n2)ktr ≥ kρ⊗n− tr1...n2 σ(n+n 2)
ktr. (7)
Using the finite quantum de Finetti theorem (Theorem 2), there exists a measure µ such that tr1...n2σ(n+n 2) = Z µ(dτ )τ⊗n+ X n, (8)
where kXnktr≤ 2dn+nn2. Thus using (7), if we prove that
kρ⊗n− Z
µ(dτ )τ⊗n+ Xn
ktr (9)
tends to 1, as n goes to infinity, we are done.
Consider an informationally complete POVM on HAand HB, and by taking their pairwise
tensor product extend them to an informationally complete POVM on H. Then apply quantum state tomography on (n − 1) copies of ρ in order to obtain an approximation of this state. To be more precise, let {Mi} be the resulting informationally complete POVM
on H. So for a sequence of outcomes (Ml1, . . . , Ml(n−1)) we get to the approximation
X i ri n − 1M ∗ i, (10)
where ri is the number of repetitions of Mi in (Ml1, . . . , Ml(n−1)).
We say that (Ml1, . . . , Ml(n−1)) is a good sequence if its corresponding estimation belongs
to Bǫ/3(ρ), the ball of radios ǫ/3 in trace distance around ρ. Let Gn−1 be the sum of Ml1 ⊗
· · · ⊗ Ml(n−1) over good sequences (Ml1, . . . , Ml(n−1)). Therefore, by the law of large numbers
[28], tr(Gn−1ρ⊗(n−1)) → 1 as n goes to infinity. Also for any τ far from ρ, tr(G
n−1τ⊗(n−1))
tends to zero.
Note that Gn−1≤ I. Thus kρ⊗n− Z µ(dτ )τ⊗n+ X n ktr≥ tr (I ⊗ Gn−1· ρ⊗n) − tr (I ⊗ Gn−1) · Z µ(dτ )τ⊗n+ X n .
Since tr (I ⊗ Gn−1· ρ⊗n) → 1, if we prove tr(I ⊗ Gn−1) · ( Z µ(dτ )τ⊗n+ Xn) → 0, we conclude that (9) tends to 1.
Now suppose that we perform quantum state tomography on R µ(dτ )τ⊗n+ X
n. By (8),
this state is not entangled. Moreover, since we can apply tomography locally (see Section II B ), by starting from a separable state, the outcome of the process is always separable as well. Assuming that we get a good sequence in the process the outcome is equal to
Z
µ(dτ ) tr[Gn−1τ⊗(n−1)] τ + eX
n, (11)
where k eXnktr ≤ kXnktr ≤ 2dn+nn2. As a result, this state is separable. (Here we assume
that (11) is non-zero because otherwise there is nothing to prove.) Let Yn = Z τ /∈Bǫ/2(ρ) µ(dτ )tr[Gn−1τ⊗(n−1)] τ + eX n, and cn= Z τ ∈Bǫ/2(ρ) µ(dτ )tr[Gn−1τ⊗(n−1)].
By the law of large numbers, there exists δn such that for any τ /∈ Bǫ/2(ρ) we have
tr[Gn−1τ⊗(n−1)] ≤ δ n,
and δn→ 0 as n goes to infinity. Therefore, kYnktr≤ δn+ 2dn+nn2.
As we mentioned, the state
e τ = 1 cn+ tr(Yn) "Z τ ∈Bǫ/2(ρ) µ(dτ )tr[Gn−1τ⊗(n−1)] τ + Y n #
is separable. On the other hand, by definition
e ρ = 1 cn Z τ ∈Bǫ/2(ρ) µ(dτ )tr[Gn−1τ⊗(n−1)] τ
is in the ǫ/2-neighborhood of ρ. Then by (6) we have
ǫ ≤ kρ − eτ ktr ≤ c cn n+ tr(Yn)kρ − eρktr + |tr(Yn)| cn+ tr(Yn)kρktr + 1 cn+ tr(Yn)kYnktr ≤ c cn n+ tr(Yn)· ǫ 2+ 2 cn+ tr(Yn)kYnktr . Thus ǫcn+ ǫtr(Yn) ≤ ǫ 2cn+ 2kYnktr,
and then cn ≤ 2(2 + ǫ) ǫ kYnktr ≤ 6ǫ −1(δ n+ 2d n n + n2).
Putting everything together we find that
tr(I ⊗ Gn−1) · ( Z µ(dτ )τ⊗n+ X n)] = tr Z τ ∈Bǫ/2(ρ) µ(dτ )tr[Gn−1τ⊗(n−1)] τ + Y n ≤ cn+ kYnktr ≤ (6ǫ−1+ 1) · (δ n+ 2d n n + n2), which gives tr(I ⊗ Gn−1) · ( Z µ(dτ )τ⊗n+ Xn)] → 0,
as n goes to infinity. We are done.
IV. GEOMETRY OF THE SET OF SEPARABLE STATES
Theorem 1 tells us that estimating the distance of a bipartite state from separable state by the distance from PPT states is not a good approximation. However, one may expect that the set of PPT states is an approximation of the set of separable states from a geometrical point of view. For instance, two spheres centered at origin with radiuses 1 and 2 are far from each other, while they have the same geometric properties up to a scaler factor. In the following theorem we bring an evidence that this is not the case for the set of separable states relative to PPT states.
By Theorem 1 the maximum distance of a PPT state from the boundary of the set of separable states is close to 1. We can think of this problem in another direction. What is the maximum distance of a state on the boundary of separable states from the boundary of PPT states? To get an intuition on this problem, we can think of the unit sphere centered at origin in Rn, and the cube with vertices (±1, . . . , ±1). It is easy to see that the distance
of any point on the sphere from points of the cube is less than 2. However, the distance of (1, . . . , 1) from sphere is√n − 1. It is because sphere and cube have totally different shapes. Theorem 3 Assume that H = HA ⊗ HB, and dim HA = dim HB = d. Then for any separable state ρ acting on H there exists a state σ on the boundary of the set of PPT states such that kρ − σktr ≤ √1
d.
Proof: Let σ be an arbitrary PPT state, and Φ(d) be the maximally entangled state defined in (3). Then the fidelity of σ and Φ(d) is
F (σ, Φ(d)) = [tr σ Φ(d)]1/2 = [tr σTB Φ(d)TB]1/2 = [tr σTB(1 dI − 2 d X i<j |φijihφij|)]1/2,
where |φiji is defined in (4). Thus using the fact that σTB is a density matrix and then
σTB ≤ I, we obtain
F (σ, Φ(d)) ≤ √1 d.
Therefore, by the well-known inequality between fidelity and trace distance (see [29] page 416) we have kσ − Φ(d)ktr≥ 1 − F (σ, Φ(d)) ≥ 1 − 1 √ d. (12)
Now let ρ be an arbitrary separable state. Define ρt= (1−t)ρ+tΦ(d). ρ0 = ρ is separable
and then PPT, and ρ1 = Φ(d). So there exists 0 ≤ c ≤ 1 such that ρc is on the boundary of
PPT states. Then we have
kρ − ρcktr= kρ − Φ(d)ktr− kρc− Φ(d)ktr ≤ 1 − (1 − 1 √ d) = 1 √ d, where in the last inequality we use (12).
This theorem together with Theorem 1 say that considering the sets of separable states and PPT states in the trace-norm space, they have completely different shapes. However, due to Dvoretzky’s theorem (see for example [30]) we know that for every convex set, its intersections with most hyperplanes of certain dimension are close to Euclidean ball in shape. This means that the set of separable states and PPT states have the same geometry if we consider them in Euclidean space and restrict them to sections of certain dimension.
V. GENERALIZATION TO OTHER SEPARABILITY CRITERIA
According to Theorem 1, if the dimension of the space is large enough, there exists a PPT state far from separable states. Our candidate for such a state is ρ⊗n, where ρ is an
entangled PPT state, and in the proof the only property of the set of PPT states that we use, is that this set is closed under tensor product. Therefore, the same argument as in the proof of Theorem 1, gives us the following general theorem.
Theorem 4 Assume that C is a necessary but not sufficient separability criterion such that if ρ and σ satisfy C, then ρ ⊗ σ satisfies C as well. Then for any ε > 0 there exists a state ρ that satisfies C, and whose trace distance from separable states is at least 1 − ε.
Proof: Let ρ be an entangled state which satisfies C. Then ρ⊗n satisfies C as well, and by
the proof of Theorem 1, the trace distance of ρ⊗n from separable states tends to 1 as n goes
to infinity.
In the following theorem we prove that all separability criteria mentioned in Section II satisfy the assumption of Theorem 4.
Theorem 5 For all separability criteria mentioned in Section II there exists an entangled state which passes the test while it is arbitrarily far, in trace distance, from separable states.
Proof: By Theorem 4 it is sufficient to prove that those separability criteria are closed under tensor product.
• Reduction criterion: Let X, Y, Z and W be positive semi-definite matrices such that X ≥ Y and Z ≥ W . Then (X − Y ) ⊗ (Z + W ) and (X + Y ) ⊗ (Z − W ) are positive semi-definite. Therefore X ⊗ Z − Y ⊗ W = 1[(X − Y ) ⊗ (Z + W ) + (X + Y ) ⊗ (Z − W )]
is positive semi-definite. It means that if X ≥ Y and Z ≥ W , then X ⊗ Z ≥ Y ⊗ U. Now assume that ρAB and σA′B′ pass the reduction criterion. Therefore ρA⊗ I ≥ ρAB
and σA′⊗I ≥ σA′B′, and then ρA⊗σA′⊗I ≥ ρAB⊗σA′B′, which means that ρAB⊗σA′B′
satisfies the reduction criterion.
• Entropic criterion: It follows easily from Sα(ρ ⊗ σ) = Sα(ρ) + Sα(σ).
• Majorization criterion: x ≺ y if and only if there exists a doubly-stochastic matrix D (a matrix all of whose entries are positive, and the sum of entries on any row and column is equal to 1) such that x = Dy (see [29] page 575). Therefore, if x ≺ y and x′ ≺ y′,
there exist D and D′ such that x = Dy and x′ = D′y′. Hence x ⊗ x′ = (D ⊗ D′)(y ⊗ y′)
and then x ⊗ x′ ≺ y ⊗ y′. The proof follows easily using this property.
• Cross norm criterion: Using v(X ⊗ X′) = v(X) ⊗ v(X′) we have U((X ⊗ X′) ⊗ (Y ⊗
Y′)) = U(X ⊗ Y ) ⊗ U(X′⊗ Y′). The proof follows from this equation.
• Symmetric extension criterion: If ρ(k) and σ(k) are symmetric extensions of ρ and σ to
k copies respectively, then ρ(k)⊗ σ(k) is a symmetric extension of ρ ⊗ σ to k copies.
VI. CONCLUSION
We have proved that for any separability criterion that is closed under tensor product, the set of states that pass the test, is not a good approximation of the set of separable states. For the special case of positive partial transpose test, we have shown that the sets of PPT states and separable states have totally different shapes. A problem that arises naturally is to find a separability criterion which is not weaker than the known ones, and also is not closed under tensor product. Finding such a separability test may clarify the complexity of the separability problem: is it NP-hard to decide whether there exists a separable state whose trace distance from a given state is less than a given constant c?
Acknowledgements. Authors are grateful to Karol ˙Zyczkowski and Stanis law J. Szarek for providing some background about the comparison of the volume of the sets of separable states and PPT states. SB is also thankful to Barbara Terhal for useful discussions.
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